Download 9-Number Systems

Computer Science 111
Fundamentals of Programming I
Number Systems
Numeric Types: int
• int is used for integers
• In many languages, the range of int is -231 through
231 - 1
• In Python, an integer’s magnitude is limited only by
the computer’s memory
Why
31
2 -
1?
• Numbers are stored in computer memory as patterns of
voltage levels
• Just two voltage levels, ON and OFF, are significant on a
digital computer
• ON and OFF represent the digits 1 and 0 of the binary
number system
• All numbers are represented in binary in a digital computer
Computer Memory
0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1
0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1
0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1
0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1
…
0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1
Memory might have billions of cells that support
the storage of trillions of binary digits or bits of
information
Each cell in this memory has room for 32 bits
Bits and Bytes
byte
byte
byte
byte
0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1
A byte is 8 bits
In some languages, int uses 4 bytes
The magnitude and the sign (+/-) of the number are
determined by the binary representation
Decimal and Binary
• Decimal numbers use the 10 decimal digits and a base of
10
• Binary numbers use the binary digits 0 and 1 and a base of
2
• The base is often provided as a subscript to indicate the
type of system, as in 304210 and 110111102
• Thus, 110110 is very different from 11012
Positional Notation
• Number systems are positional, so the magnitude of the number
depends on the base and the position of the digits in the number
• Each position represents a power of the number’s base
• For example, in a 3-digit decimal number, the three positions represent
the number of hundreds (102), tens (101), and ones (100)
•
•
•
•
342 = 3 * 102 + 4 * 101 + 2 * 100
= 3 * 100 + 4 * 10 + 2 * 1
= 300 + 40 + 2
= 342
Positional Notation: Binary
• The base is now 2 and the only digits are 0 and 1
• Each position represents a power of 2
• For example, in a 4-digit binary number, the four positions represent
the number of eights (23), fours (22), twos (21), and ones (10)
•
•
•
•
1101 = 1 * 23 + 1 * 22 + 0 * 21 + 1 * 20
=1*8+1*4+0*2+1*1
=8+4+0+1
= 13
An Algorithm for Binary to
Decimal Conversion
# Input: A string of 1 or more binary digits
# Output: The integer represented by the string
binary = input("Enter a binary number: ")
decimal = 0
exponent = len(binary) – 1
for digit in binary:
decimal = decimal + int(digit) * 2 ** exponent
exponent = exponent – 1
print("The integer value is", decimal)
The len function returns the number of
characters in a string
The for loop visits each character in a string
Counting in Binary
Binary
Magnitude
0
0
1
1
10
2
11
3
100
4
101
5
110
6
111
7
1000
8
21
Each power of 2 in
binary is a 1 followed by
the number of 0s equal to
the exponent
22
23
What is the magnitude of
10000002?
Counting in Binary
Binary
Magnitude
0
0
1
1
10
2
11
3
100
4
101
5
110
6
111
7
1000
8
21 - 1
22 - 1
23 - 1
Each number with only
1s equals one less than
the power of 2 whose
exponent is that number
of 1s
What is the magnitude of
11112?
Limits of Magnitude - Unsigned ints
• Unsigned integers are the non-negative integers
• The largest unsigned integer that can be
represented using N bits is 2N - 1 (all bits are 1s)
• Thus, the largest unsigned integer stored in 32 bits
is 232 - 1
Limits of Magnitude - Signed ints
• Signed integers include negative and positive integers and
0
• Part of the memory (one bit) must be reserved to represent
the number’s sign somehow
• For each bit unavailable, you must subtract 1 from the
exponent (2N-1) of the number’s magnitude
• Thus, the largest positive signed integer stored in 32 bits is
231 - 1
Twos Complement Notation
• Positive numbers have 0 in the leftmost bit, negative
numbers have 1 in the leftmost bit
• To compute a negative number’s magnitude,
– Invert all the bits
– Add 1 to the result
– Use the conversion algorithm
• To represent a negative number,
– Translate the magnitude to an unsigned binary number
– Invert all the bits
– Add 1 to the result
Convert Decimal to Binary
• Start with an integer, N, and an empty string, S
• Assume that N > 0
• While N > 0:
– Compute the remainder of dividing N by 2 (will be 0 or 1)
– Prepend the remainder’s digit to S
– Reset N to the quotient of N and 2
An Algorithm for Decimal to
Binary Conversion
# Input: An integer > 0
# Output: A string representing the integer in base 2
n = int(input("Enter an integer greater than 0: "))
binary = ''
while n > 0:
rem = n % 2
binary = str(rem) + binary
n = n // 2
print(binary)
Here we want the quotient and the remainder, not
exact division!
Numeric Types: float
• Uses 8 bytes to represent floating-point numbers
that range from +10308.25 through -10308.25
• Default printing is up to 16 digits of precision
• But actual precision seems to extend even further try format string with 60 places
Problems with Float
• Some numbers, such as 0.1, cannot be
represented exactly, due to round-off errors
• This can cause errors when two floats are
compared for equality
Example: Add 0.1 Ten Times
sum = 0.0
for i in range(10):
sum = sum + 0.1
print(sum)
print(sum < 1.0)
# Displays 1.0
# Displays True!
Convert dollars and cents to pennies (ints) in
critical financial calculations!
Use int or float?
• In general, arithmetic operations are faster
with integers than with floats
• Integers (regular ones) require less memory
• Use float when it’s convenient to do so
and you’re not worried about losing
numeric information
Reading For Monday
Finish Chapter 4